Opuscula Math. 44, no. 2 (2024), 285-296
https://doi.org/10.7494/OpMath.2024.44.2.285

 
Opuscula Mathematica

Weak signed Roman k-domination in digraphs

Lutz Volkmann

Abstract. Let \(k\geq 1\) be an integer, and let \(D\) be a finite and simple digraph with vertex set \(V(D)\). A weak signed Roman \(k\)-dominating function (WSRkDF) on a digraph \(D\) is a function \(f \colon V(D)\rightarrow \{-1,1,2\}\) satisfying the condition that \(\sum_{x \in N^-[v]}f(x)\geq k\) for each \(v\in V(D)\), where \(N^-[v]\) consists of \(v\) and all vertices of \(D\) from which arcs go into \(v\). The weight of a WSRkDF \(f\) is \(w(f)=\sum_{v\in V(D)}f(v)\). The weak signed Roman \(k\)-domination number \(\gamma_{wsR}^k(D)\) is the minimum weight of a WSRkDF on \(D\). In this paper we initiate the study of the weak signed Roman \(k\)-domination number of digraphs, and we present different bounds on \(\gamma_{wsR}^k(D)\). In addition, we determine the weak signed Roman \(k\)-domination number of some classes of digraphs. Some of our results are extensions of well-known properties of the weak signed Roman domination number \(\gamma_{wsR}(D)=\gamma_{wsR}^1(D)\) and the signed Roman \(k\)-domination number \(\gamma_{sR}^k(D).\)

Keywords: digraph, weak signed Roman \(k\)-dominating function, weak signed Roman \(k\)-domination number, signed Roman \(k\)-dominating function, signed Roman \(k\)-domination number.

Mathematics Subject Classification: 05C20, 05C69.

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  • Lutz Volkmann
  • Lehrstuhl II für Mathematik, RWTH Aachen University, 52056 Aachen, Germany
  • Communicated by Dalibor Fronček.
  • Received: 2022-08-18.
  • Revised: 2023-08-22.
  • Accepted: 2023-08-23.
  • Published online: 2024-01-15.
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Cite this article as:
Lutz Volkmann, Weak signed Roman k-domination in digraphs, Opuscula Math. 44, no. 2 (2024), 285-296, https://doi.org/10.7494/OpMath.2024.44.2.285

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